In Exercises 35–38, find the exact value of the following under the given conditions:
e. cos( β/2)
sin α = -1/3, 𝝅 < α < 3𝝅/2, and cos β = -1/3, 𝝅 < β < 3𝝅/2.

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Identify the given information: \( \sin \alpha = -\frac{1}{3} \) with \( \pi < \alpha < \frac{3\pi}{2} \), and \( \cos \beta = -\frac{1}{3} \) with \( \pi < \beta < \frac{3\pi}{2} \). Both angles are in the third quadrant.

Recall that in the third quadrant, both sine and cosine values are negative, which is consistent with the given values and intervals.

Use the Pythagorean identity to find \( \cos \alpha \): \( \cos \alpha = -\sqrt{1 - \sin^2 \alpha} = -\sqrt{1 - \left(-\frac{1}{3}\right)^2} \). The negative sign is chosen because cosine is negative in the third quadrant.

Similarly, find \( \sin \beta \) using the Pythagorean identity: \( \sin \beta = -\sqrt{1 - \cos^2 \beta} = -\sqrt{1 - \left(-\frac{1}{3}\right)^2} \), since sine is also negative in the third quadrant.

To find \( \cos \frac{\beta}{2} \), use the half-angle formula: \[ \cos \frac{\beta}{2} = \pm \sqrt{\frac{1 + \cos \beta}{2}}. \] Determine the correct sign based on the quadrant where \( \frac{\beta}{2} \) lies, considering \( \pi < \beta < \frac{3\pi}{2} \).

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